This invention relates to a method of forming free-form curves and method of forming free-form surfaces, and more particularly is applicable to a designing device using the technique of CAD/CAM (computer aided design/computer aided manufacturing) for example.
In the case where the technique of CAD is used to design the shape of an object having free-form surface (geometric modeling), the designer usually designates a plurality of points (referred to as nodal points) in a three-dimensional space which are passed through by the curved surface. A surface represented by a so-called wire-frame is formed by causing a desired vector function to calculate a boundary curve network which connects the designated nodal points. Thereby, a number of framing spaces surrounded by the boundary curves may be formed (such processing is referred to as framing process).
The boundary curve network formed such framing process by itself represents a generalized shape of a design intended by the designer. A free-form surface (referring to one which cannot be defined by a quadratic function) designed as a whole by the designer may be generated, if it is possible by interpolation to obtain a curved surface which may be represented by a predetermined vector function using the boundary curves surrounding the respective framing spaces. Here, the curved surface pasted on each framing space forms a fundamental element for constructing the total curved surface, and it is referred to as a patch.
In order to give a more natural appearance of the shape to the generated free-form surface as a whole, a method of forming a free-form curve (Japanese Patent Application No.60-277448) has been proposed, in which, for two framing spaces adjoining each other with an interposing common boundary, the control side vectors around the common boundary is redetermined so as to paste a patch which satisfies the condition of continuity of tangential planes at the common boundary.
As shown in FIG. 1, the principle of such free-form surface forming method is that: patch vector S(u,v)1 and patch vector S(u,v)2 to be pasted onto quadrilateral framing spaces are represented by a vector function S(u,v) consisting of a third order Bezier expression; in order to smoothly connect the two patch vectors S(u,v)1 and vector S(u,v)2, control side vectors, vector a1, vector a2 and vector c1, vector c2 are determined so that the condition of continuity of tangential planes is satisfied at the common boundary COM of the adjoining patch vectors S(u,v)1 and S(u,v)2 on the basis of the nodal points, vector P(00), vector P(30)1, vector P(33)1, vector P(03), vector P(33)2 and vector P(30)2, which are given by the framing process; and the control point vectors, vector P(11)1, vector P(12)1, vector P(11)2 and vector P(12)2, are redetermined by these control side vectors.
As a result of applying such technique also to other common boundaries, the patch vectors, vector S(u,v)1 and vector S(u,v)2, may be smoothly connected to adjoining patches in accordance with the condition of continuity of tangential planes.
Here, the vector function vector S(u,v) formed of a third order Bezier expression is represented using parameters u and v in the u direction and the v direction and shift operation E and F by the following formula: EQU S(u,v)=(1-u+uE).sup.3 (1-v+vF).sup.3 P(00) (1)
and is related to the control point vectors P(ij) as follows: EQU E.multidot.P(ij)=P(i+1j) (i,j=0,1,2) (2) EQU F.multidot.P(ij)=P(ij+1) (i,j=0,1,2) (3) EQU 0.ltoreq.u.ltoreq.1 (4) EQU 0.ltoreq.v.ltoreq.1 (5)
Further, a tangential plane refers to the plane formed by the tangential vectors in the u direction and the v direction at each point on the common boundary. For example, the condition of continuity of tangential planes is satisfied with respect to the common boundary COM12 of FIG. 1 when the tangential planes of the patch vectors, vector S(u,v)1 and vector S(u,v)2, are identical to each other.
According to this method, designing is readily possible of the shape of an object such as one with a surface geometry changing smoothly as a whole exactly as intended by the designer, which has been practically difficult to be designed by a conventional designing method.
With such a designing device, it is presumed that the operability of the designing device may be improved and it is convenient if the contour of the object to be designed may be inputted as a series of points.
To this end, it is necessary to form a wire-frame model in a manner of connecting the input series of points.
Further, the free-form curves forming the wire-frame model must be represented by Bezier curves.
Furthermore, the free-form surface must be generated with respect to the generated wire-frame model so that it passes through a group of points.
In addition, it is presumably convenient if the geometric shape of thus generated patch is corrected using a curved surface such as of circular arc.
If a change in the shape of a curved surface is possible in this manner, the size of a patch may be set over again as required and it is also possible to set the framing space itself over again.